If we lift the restriction \(r=s=1\)  then in the jump matrix of the \(\mathcal {X}\) -RHP
the functions \(\phi (z)\)  and \(w(z)\)  should be replaced by \(z^{1-r}\phi (z)\) 
and \(z^{1-s}w(z)\) , respectively. This, in turn would raise a serious question about solvabilty
of the \(\Lambda \) -RHP. Indeed, for instance, we would not be able to define the function \(\alpha \)  in (REF ) and
hence to factorize the jump matrix (REF ). The way out of this difficulty could be the
use of the relation between the determinants \(D_n(\phi ,w;r,s)\)  with different values of \(r\) , \(s\)  or both. Such relations
are well known in the pure Toeplitz case (for example see [1]}). However, for general Toeplitz+Hankel determinants they are yet to be found.
Another way to approach the problem could be to develop the so-called Bäklund–Schlesinger transformations
of the \(\mathcal {X}\) -RHP itself. That is, the transformations of the form,
\(\mathcal {X}(z) \mapsto R(z)\mathcal {X}(z),\) 
